Differentiation under Integral Sign

Q.

$\frac{d}{dx}(\log (secx+\tan x\left)\right)=$

1. $\cos ecx$

2. $secx$

3. $\tan x$

4. $\cos x$

Q.

If $\underset{x\to 0}{lim}\frac{2a\sin x-\sin 2x}{{\tan }^{3}x}$ exists and is equal to $1$, then the value of $a$ is:

1. $2$

2. $1$

3. $0$

4. $-1$

Q. Let $f\left(x\right)$ be a non-negative continuous function such that the area bounded by the curve $y=f\left(x\right),$ $x$-axis and the ordinates $x=\frac{\pi }{4}$ and $x=\beta >\frac{\pi }{4}$ is $\beta \sin \beta +\frac{\pi }{4}\cos \beta +\sqrt{2}\beta .$ Then $f^{\prime }\left(\frac{\pi }{2}\right)$ is

1. $(\frac{\pi }{2}-\sqrt{2}-1)$
2. $(\frac{\pi }{4}+\sqrt{2}-1)$
3. $-\frac{\pi }{2}$
4. $(1-\frac{\pi }{4}-\sqrt{2})$

Q. 12. f (x+y) + f (x-y) = 2f(x)f (y) and f (alpha) = -1 . Find the period of f

Q. If $f\left(x\right)$ is differentiable and ${\int }_0^{t^2}xf\left(x\right)dx=\frac{2}{5}{t}^5,\text{then}f\left(\frac{4}{25}\right)$ equals

1. $\frac{2}{5}$
2. $-\frac{5}{2}$
3. $1$
4. $\frac{5}{2}$

Q.

If f(x) = x² – x + 1; g(x) = 7x – 3, be two real functions then (f + g)(3) is

1. 18

2. 3

3. 25

4. 7

Q. A function y = f(x) satisfies the condition $f\left(x\right)sinx+f\left(x\right)cosx=1$, $f\left(x\right)$ being bounded when $x\to 0$. If $|=\underset{0}{\overset{\frac{\pi }{2}}{\int }}f\left(x\right)dx$ then

1. $\frac{\pi }{2}<|<\frac{{\pi }^2}{4}$
2. $\frac{\pi }{4}<|<\frac{{\pi }^2}{2}$
3. $1<|<\frac{\pi }{2}$
4. $0<|<1$

Q. If f is a continuous function then, ${\int }_0^{2a}f\left(x\right)dx={\int }_0^{a}f\left(x\right)dx+{\int }_0^{2a}f(2a-x)dx.$

1. True
2. False

Q. 45. The largest Set of real values of x for which f(x)=(x+2)(5-x) - 1/x-4 is a real function is 1)(2, 5] 2)[3, 4]

Q. (i) $\frac{2}{x}$

(ii) $\frac{1}{\sqrt{x}}$

(iii) $\frac{1}{x^3}$

(iv) $\frac{x^2+1}{x}$

(v) $\frac{x^2-1}{x}$

(vi) $\frac{x+1}{x+2}$

(vii) $\frac{x+2}{3x+5}$

(viii) k xⁿ

(ix) $\frac{1}{\sqrt{3-x}}$

(x) x² + x + 3

(xi) (x + 2)³

(xii) x³ + 4x² + 3x + 2

(xiii) (x² + 1) (x − 5)

(xiv) $\sqrt{2x^2+1}$

(xv) $\frac{2x+3}{x-2}$

Q. Consider $f\left(x\right)=1+\underset{0}{\overset{2x}{\int }}\frac{e^{\frac{t}{2}}\cdot f\left(\frac{t}{2}\right)\left(2t\right)}{\sqrt{16-t^4}}dt-\underset{x}{\overset{0}{\int }}f\left(t\right)\cdot e^t{\sin }^{-1}\left(t^2\right)dt$ and $h\left(x\right)=\sin \left(e^{-x}\ln \right(f\left(x\right)\left)\right)$.
Then the range of $y=h\left(x\right)+4x+5$ is

1. $[1,\infty )$
2. $(1,\infty )$
3. $[2,10]$
4. $(2,10)$

Q. $\underset{x\to 0}{\lim }\frac{x}{\tan x}\mathrm{is}$

(a) 0
(b) 1
(c) 4
(d) not defined

Q. Let $f\left(x\right)$ be a continuous and not a constant function of all $x$ in its domain, such that
$\left(f\right(x){)}^2=\underset{0}{\overset{x}{\int }}f\left(t\right)\frac{4}{\sin 2t-4{\sin }^{2}t+4}dt$ and $f\left(0\right)=0,$ then

1. $f\left(\frac{3\pi }{4}\right)=\log \left(\frac{1}{2}\right)$
2. $f\left(\frac{\pi }{4}\right)=\log \left(\frac{3}{2}\right)$
3. $f\left(\frac{\pi }{4}\right)=\log \left(\frac{5}{4}\right)$
4. $f\left(\frac{\pi }{2}\right)=2$

Q. If $I=\underset{0}{\overset{x}{\int }}\left[\cos t\right]dt$, where $x\in \left[\right(4n+1)\frac{\pi }{2},(4n+3\left)\frac{\pi }{2}\right],n\in N$ and $[\cdot ]$ represents greatest integer function, then the value of $I$ is

1. $\frac{\pi }{2}(2n-1)-2x$
2. $\frac{\pi }{2}(2n-1)+x$
3. $\frac{\pi }{2}(2n+1)-x$
4. $\frac{\pi }{2}(2n+1)+x$

Q. Let $f\left(x\right)$ satisfies the relation $f\left(x\right)=\cos x-\underset{0}{\overset{x}{\int }}{f}^{\prime }\left(t\right)(2\cos t+{\cos }^{2}t)dt$. Then, which of the following is/are correct?

1. The maximum value of $f\left(x\right)$ is $1$.
2. The maximum value of $f\left(x\right)$ is $-2$.
3. $f\left(x\right)$ is a periodic function with fundamental period $2\pi$.
4. $f\left(x\right)$ is a bounded function.

Q. If $\underset{0}{\overset{x}{\int }}f\left(t\right)dt=x^2+\underset{x}{\overset{1}{\int }}{t}^{2}f\left(t\right)dt$, then $f^{\prime }\left(\frac{1}{2}\right)$ is:

1. $\frac{18}{25}$
2. $\frac{4}{5}$
3. $\frac{24}{25}$
4. $\frac{6}{25}$

Q. The intercepts on $x$-axis made by tangents to curve, $y=\underset{0}{\overset{x}{\int }}|t|dt,x\in R$, which are parallel to the line $y=2x$, are equal to:

1. $\pm 1$
2. $\pm 2$
3. $\pm 3$
4. $\pm 4$

Q. Let f be a given function continuous and derivable for all x and satisfying the relation f(x+y). f(x-y) = f²(x). If f(0) ≠0 find f(x)

Q.

if the ratio of the roots of ax²+bx+c=0 is equal to the roots of the equation x²+x+1=0 then a, b, c are in

a)A.P

b)GP

c)HP

d)none